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Theorem invghm 15455
Description: The inversion map is a group automorphism if and only if the group is abelian. (In general it is only a group homomorphism into the opposite group, but in an abelian group the opposite group coincides with the group itself.) (Contributed by Mario Carneiro, 4-May-2015.)
Hypotheses
Ref Expression
invghm.b  |-  B  =  ( Base `  G
)
invghm.m  |-  I  =  ( inv g `  G )
Assertion
Ref Expression
invghm  |-  ( G  e.  Abel  <->  I  e.  ( G  GrpHom  G ) )

Proof of Theorem invghm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 invghm.b . . 3  |-  B  =  ( Base `  G
)
2 eqid 2438 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
3 ablgrp 15419 . . 3  |-  ( G  e.  Abel  ->  G  e. 
Grp )
4 invghm.m . . . . 5  |-  I  =  ( inv g `  G )
51, 4grpinvf 14851 . . . 4  |-  ( G  e.  Grp  ->  I : B --> B )
63, 5syl 16 . . 3  |-  ( G  e.  Abel  ->  I : B --> B )
71, 2, 4ablinvadd 15436 . . . 4  |-  ( ( G  e.  Abel  /\  x  e.  B  /\  y  e.  B )  ->  (
I `  ( x
( +g  `  G ) y ) )  =  ( ( I `  x ) ( +g  `  G ) ( I `
 y ) ) )
873expb 1155 . . 3  |-  ( ( G  e.  Abel  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  ( x ( +g  `  G ) y ) )  =  ( ( I `  x ) ( +g  `  G
) ( I `  y ) ) )
91, 1, 2, 2, 3, 3, 6, 8isghmd 15017 . 2  |-  ( G  e.  Abel  ->  I  e.  ( G  GrpHom  G ) )
10 ghmgrp1 15010 . . 3  |-  ( I  e.  ( G  GrpHom  G )  ->  G  e.  Grp )
1110adantr 453 . . . . . . . 8  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  G  e.  Grp )
12 simprr 735 . . . . . . . 8  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  y  e.  B )
13 simprl 734 . . . . . . . 8  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  x  e.  B )
141, 2, 4grpinvadd 14869 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  y  e.  B  /\  x  e.  B )  ->  ( I `  (
y ( +g  `  G
) x ) )  =  ( ( I `
 x ) ( +g  `  G ) ( I `  y
) ) )
1511, 12, 13, 14syl3anc 1185 . . . . . . 7  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  ( y ( +g  `  G ) x ) )  =  ( ( I `  x ) ( +g  `  G
) ( I `  y ) ) )
1615fveq2d 5734 . . . . . 6  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  ( I `  (
y ( +g  `  G
) x ) ) )  =  ( I `
 ( ( I `
 x ) ( +g  `  G ) ( I `  y
) ) ) )
17 simpl 445 . . . . . . 7  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  I  e.  ( G  GrpHom  G ) )
181, 4grpinvcl 14852 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  x  e.  B )  ->  ( I `  x
)  e.  B )
1911, 13, 18syl2anc 644 . . . . . . 7  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  x )  e.  B
)
201, 4grpinvcl 14852 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  y  e.  B )  ->  ( I `  y
)  e.  B )
2111, 12, 20syl2anc 644 . . . . . . 7  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  y )  e.  B
)
221, 2, 2ghmlin 15013 . . . . . . 7  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
I `  x )  e.  B  /\  (
I `  y )  e.  B )  ->  (
I `  ( (
I `  x )
( +g  `  G ) ( I `  y
) ) )  =  ( ( I `  ( I `  x
) ) ( +g  `  G ) ( I `
 ( I `  y ) ) ) )
2317, 19, 21, 22syl3anc 1185 . . . . . 6  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  ( ( I `  x ) ( +g  `  G ) ( I `
 y ) ) )  =  ( ( I `  ( I `
 x ) ) ( +g  `  G
) ( I `  ( I `  y
) ) ) )
241, 4grpinvinv 14860 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  x  e.  B )  ->  ( I `  (
I `  x )
)  =  x )
2511, 13, 24syl2anc 644 . . . . . . 7  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  ( I `  x
) )  =  x )
261, 4grpinvinv 14860 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  y  e.  B )  ->  ( I `  (
I `  y )
)  =  y )
2711, 12, 26syl2anc 644 . . . . . . 7  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  ( I `  y
) )  =  y )
2825, 27oveq12d 6101 . . . . . 6  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( (
I `  ( I `  x ) ) ( +g  `  G ) ( I `  (
I `  y )
) )  =  ( x ( +g  `  G
) y ) )
2916, 23, 283eqtrd 2474 . . . . 5  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  ( I `  (
y ( +g  `  G
) x ) ) )  =  ( x ( +g  `  G
) y ) )
301, 2grpcl 14820 . . . . . . 7  |-  ( ( G  e.  Grp  /\  y  e.  B  /\  x  e.  B )  ->  ( y ( +g  `  G ) x )  e.  B )
3111, 12, 13, 30syl3anc 1185 . . . . . 6  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( y
( +g  `  G ) x )  e.  B
)
321, 4grpinvinv 14860 . . . . . 6  |-  ( ( G  e.  Grp  /\  ( y ( +g  `  G ) x )  e.  B )  -> 
( I `  (
I `  ( y
( +g  `  G ) x ) ) )  =  ( y ( +g  `  G ) x ) )
3311, 31, 32syl2anc 644 . . . . 5  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  ( I `  (
y ( +g  `  G
) x ) ) )  =  ( y ( +g  `  G
) x ) )
3429, 33eqtr3d 2472 . . . 4  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( x
( +g  `  G ) y )  =  ( y ( +g  `  G
) x ) )
3534ralrimivva 2800 . . 3  |-  ( I  e.  ( G  GrpHom  G )  ->  A. x  e.  B  A. y  e.  B  ( x
( +g  `  G ) y )  =  ( y ( +g  `  G
) x ) )
361, 2isabl2 15422 . . 3  |-  ( G  e.  Abel  <->  ( G  e. 
Grp  /\  A. x  e.  B  A. y  e.  B  ( x
( +g  `  G ) y )  =  ( y ( +g  `  G
) x ) ) )
3710, 35, 36sylanbrc 647 . 2  |-  ( I  e.  ( G  GrpHom  G )  ->  G  e.  Abel )
389, 37impbii 182 1  |-  ( G  e.  Abel  <->  I  e.  ( G  GrpHom  G ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   -->wf 5452   ` cfv 5456  (class class class)co 6083   Basecbs 13471   +g cplusg 13531   Grpcgrp 14687   inv gcminusg 14688    GrpHom cghm 15005   Abelcabel 15415
This theorem is referenced by:  gsuminv  15543  invlmhm  16120  tsmsinv  18179
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-riota 6551  df-0g 13729  df-mnd 14692  df-grp 14814  df-minusg 14815  df-ghm 15006  df-cmn 15416  df-abl 15417
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